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Still looking for a special present for a special person? Then what about a Gömböc: this strange egg-like shape wriggles around with an apparent will of its own and, even better, barely exists at all.

A Gömböc is a 3D convex shape with one stable and one unstable equilibrium point. The reason it wriggles around when you put it down is that it's "trying to" rest on its stable equilibrium point, and since there's only one of them it takes a little dance to find it. for other objects this doesn't happen because they have more stable equilibrium points, so all they need to do to find one when you put them down in an unstable position is fall over. If you change a Gömböc even only a tiny bit, you'll create extra equilibrium points or stop it from being convex, so you'll stop it from being a Gömböc. This is why Gömböcs teeter on the brink of existence — the smallest change and they're gone.

For a very long time people believed that Gömböcs didn't exist, but recently the Hungarian mathematicians Gábor Domokos and Péter Várkonyi proved that they did (find out more here). You can purchase your very own Gömböc on the Gömböc website (though be warned, it's a little pricey!).

If you're feeling a bit weary in the run up to Christmas (I know we are!) we thought you might like to put your feet up and have a cup of tea instead of whatever it is you are working on. You can relax because every thing you've ever produced or will produce is already encoded in a number known as Champernowne's constant, consisting of every positive whole number listed after the decimal point:

0.1234567891011121314151617….

This is because this number is normal, which implies it contains a copy of every finite string of numbers in its infinite decimal expansion. This includes every article we have ever written or will write, every song anyone's composed, every film shot, every report written and every spreadsheet created, translated into numerical form (as it would be when stored on a computer). Lots of more famous numbers, such as , e and are also thought to be normal. In fact although mathematicians know almost every number is normal, they have only been able to prove this is true for a handful.

Unfortunately for you, and for us as we try to finish a few more articles before Christmas, even though all the work we will ever write is already present in the digits of Champernowne's number, we aren't going to be able to head to the pub just yet. Our articles and everyone else's work are swamped by every other possible string of numbers and we're going to have to produce them the hard way after all.

The idea is beautifully explored by the Argentine author Jorge Luis Borges in his short story "The Library of Babel" (published in the anthology Fiocciones). His library contains every book that is possible to write in a given alphabet, shelved in a seemingly endless complex of connected identical rooms (it looks a bit like the strange book filled place Matthew McConaughey found himself in towards the end of the Interstellar). The librarians were initially overjoyed to discover that the library contained every possible book. But their joy soon turned to despair when they realised that virtually all the books in the library would be nonsensical, the pages randomly filled with letters. They would spend their lives journeying through the endless identical rooms in a quest to find meaning among the books, knowing they were almost certain never to find it.

Borges' writings are full of such inventive and poetic explorations of philosophical and moral ideas. We came across this beautiful story when writing about normality and randomness for our book Numericon, and ended up devouring the rest of Fiocciones and have since gone on to read many others. You can also read more about the maths behind this story in The amazing librarian on Plus.

No collection of popular maths books would be complete without a work by one of our very favourite authors, Ian Stewart. One we particularly like is 17 equations that changed the world. As the name suggests, it's about equations that have had a profound impact on humankind, from Pythagoras's theorem about right-angled triangles to the Black-Scholes equation about financial derivatives. It's a fascinating and accessible tour through some very interesting maths.

Stewart has written plenty of popular maths books, which are all well worth looking at, but there's another one we'd like to mention: Does god play dice
is a great introduction to mathematical chaos, which inspired one of us to go and do a PhD in maths — she's never looked back!

How about whiling away your time on a planet that's made entirely of beaches and where it's always Saturday afternoon? If that sounds good, then you better get your hands on the
Hitchhiker's guide to the galaxy, the indispensable companion of intergalactic travellers. We revisited Douglas Adam's cult trilogy in five parts when we were writing our book (Numericon) and sill loved it as much as when we read it for the first time.

If you're not familiar with this classic, it starts with the Earth being demolished to make way for an intergalactic superhighway and the only two survivors hitching a ride on a passing spacecraft to embark on a journey that holds surprises way more bizarre, and menacing, than Saturday afternoon planets. If you are one of the many people who do know the
Hitchhiker's guide then consider rereading it. It's fun!

The travelling salesman is one of our favourite mathematical movies. It's a tense thriller revolving around one of the most difficult open problems in maths, the P vs NP problem, and its potential to deliver the key to the world's most secret messages. Most of the movie is set in a single room, a secret government location, where four mathematicians are being debriefed as their highly classified project has been completed. An unexpected by-product of their work is a method for cracking the codes used to encrypt classified messages, giving rise to an intense debate between the mathematicians. Will their work be used for evil, by governments (or worse) to spy on all our communications and data? Would keeping it secret hamper medical advances and scientific discoveries that could be a force for good?
The result is an intelligent movie full of suspense that takes maths, as well as mathematicians, seriously. Visit the movie's website for download or instant streaming.